There are 2 separate issues here. One is what your rights are as far as being in a property that isn't in your name, and assume, one where you haven't made a financial contribution directly towards (as you say that your ex DP is paying the mortgage)
The other issue is what are the rights of your children, as dependants?
As you are probably aware, common law property is a myth and unless one makes a direct contribution to the property, then you don't have rights.
It is very unlikely that you would be kicked out with a dependant, but your DP could probably reasonably sell the property and give you enough notice.
But if (ex) DP is offering to give you a lump sum of the property, then this is generous of him, and most likely more than you could expect if you decided to fight him on it.
Further, at the moment, you have his goodwill, which is such an asset.
Get this legally side checked by a family law solicitor, but I would be gracious and accept his offer as long as it was enough to set you and child up in suitable accomodation.
Lovely that some splits are amicable